One of the most remarkable books of humankind is the Chinese I Ching, the “Book of Changes.” It was originally made up only of marks, consisting of unbroken - and broken - - lines; this fact, and possibly more importantly the respect and high veneration for the commentaries and legends surrounding the book in the most ancient times of Chinese intellectual history, allowed it to escape the great book burnings of Qin Shi Huang. In the introduction to his translation of the I Ching (Jena, Diederichs, 1924, 2 vols.), R. Wilhelm writes:

“Up to modern times, the most distinguished Chinese scholars study it. Almost all the great and important ideas that have emerged for over 3,000 years of ancient Chinese history have in part been inspired by this book, and in part have had an influence on its further clarification, so that one can reasonably state that the most mature wisdom of the millennia is worked into the I Ching. Its existence is attributed to the four holy, mythical figures, meaning that its age spans all of living history. The eight basic signs [see below] do not even have names in the Chinese language, and thus it has been assumed that they are foreign in origin. The version of the I Ching that we have today goes back to Confucius, who was approximately contemporary with Pythagoras, Buddha, and Lao Tzu (ca. 600 b.c.). Confucius meditated upon the diagrams, wrote down his thoughts, surely preserving yet older interpretations, and thus over the course of the centuries there finally emerged a great commentary, which only in modern times was separated from the tangled mass of superstitious marginalia that had crept in, and which now can be shared as a unified whole with our European thought.”

There are two basic ideas in the I Ching. One is the idea of change, i.e. the continual change of all relationships in a world of opposites: Yang = light, male, heaven, and Yin = dark, female, earth. The other idea is the ideology. Everything that happens is the image-concept, i.e. the effect of an image, an idea, in the invisible.

In these two fundamental thoughts, Confucius and Lao Tzu completely agree. Lao Tzu, in his 42nd speech, writes: “One has brought forth two, two has brought forth three,” to which one of the old Confucian commentators remarks: “These words mean: One has divided itself into Yin = the female principle and Yang = the male principle. These two have joined, and from their union, harmony emerges (as the 3rd). The exhalation of harmony (Ki-ho), however, compressing itself, has produced all beings” (Thimus, Harmonikale Symbolik I, 80 and 193). Later in his 42nd speech, Lao Tzu writes: “All beings flee from stasis and seek movement. An empty breath, connecting things, generates harmony.” Even more descriptive are the words right at the beginning of the Tao Te Ching: “The nameless Nothing is the originator of heaven and earth ... the endless Nothing can only be seen in its invisible spiritual existence, the finite Something is seen in the form of its limitation. These two opposites are one because of their primordial existence, only they are designated in different ways. Both are called depths. They are depths, twofold depths. That is the gate to all extrasensory things” (from Thimus I, 193-194).

We also learn (Thimus I, 81) that the “Tao” (inadequately translated as “understanding,” “sense,” “path,” “right way,” etc.) stands on these two principles Yang and Yin as something no longer adherent to the corporeal, purely rational, and intelligible. The Tao is also known as Chang-ti, the highest lord of heaven = God. Thus we can add the following to the beginning of the “P”:

Figure 464

in order to find these basic ideas of Chinese philosophy once again in precise harmonic correspondence. A. von Thimus had the idea of bringing this identity (I, 83 ff.), the principles Yang and Yin, for which the I Ching uses an unbroken - and broken - - line, in connection with the reciprocal partial-tone series. Namely, in the I Ching the following four images, the Se-Siang, are constructed from these lines:

--- ------

Since Confucius, in his commentaries, identifies these images with “Siang,” i.e. mutuality, reciprocity-a reciprocity from which the entire system of the I Ching hexagrams is built-the idea of the model of the reciprocal “P” suggests itself in these first four Se-Siang:

The four basic images (Se-Siang = reciprocities)

Figure 465

The “Se-Siang” are thus to be assessed as the four basic tendencies from which the system of the world comes, surrounded by the heavenly number of the “great Yang” -- and the earthly number of the “great Yin” -- and completed by the two permeating tendencies that realize the structure of the world, the “small Yang” - - and the “small Yin” ---.

From these four primal signs, the famous eight trigrams are constructed, through the addition of new lines; they are then the basis for the hexagrams of the I Ching that emerge through their permutation. These eight signs, which are not images of things, but images of tendencies of relation and change (I Ching = the Book of Changes), have the following arrangement and meaning, according to R. Wilhelm, Vol. I, p. V (see the table below).

From these diagrammatic roots, out of which the coupling of each pair yields 64 combinations, the hexagrams of the I Ching are constructed. For harmonic analysis, we choose from these the four that A. Amiot gives as the basis for the elicitation of the Chinese tone-system, in his “Mémoire sur la musique des Chinois,” in Mémoires concernant le chinois, vol. 6, Paris 1779 (from Thimus I, 86):

Thimus (I, 87 ff.) now proceeds by using for the unbroken lines of the Kien the product of a · b · c · d · e · f, or

a · b · c · d · e · f

1

and for the broken lines of the Kouen, the corresponding reciprocal term

______1______

a · b · c · d · e · f

In the ancient Hebrew book of Sepher Jezirah, Ch. 4 reads: “Two letters build 2 houses, 3 build 6 houses, 4 build 24 houses, 5 build 120 houses, 6 build 720 houses, and from there it goes on, and think what the mouth cannot speak and the ear cannot hear.” If one substitutes for a b c d e f the first numbers of the number-series as first product one obtains the factorial number 6! = 720, and for the second reciprocal product, the fraction 1/6! = 1/720 as a numeric expression for the hexagrams Kien and Kouen. Regarding the number 720, the 11th verse from the unity song of Dirghatama from the hymns ofthe Rig Veda appears to me to be relevant:

If one now sets the tone d as generator tone for 1/1 (for reasons of symmetry of accidentals, instead of 1/1c), then the above four hexagrams get the tone-values:

720/1gisˆ9

48/15bˆ2

15/48fis2

1/720as10

Kien

Weiki

Kiki

Kouen

i.e. in the “heavenly number,” 720/1, and the “earthly number, 1/720, the two imaginary poles of an “inaudible harmony” (̔αρμονία̕αφανής) and in the two ratios 48/15 and 15/48 the two poles of a scale to be projected from the imaginary into this “real” space. There are two ways of obtaining this. Either we build from the imaginary poles through the “way up” in duodecimal (octave plus fifth) intervals, thus via the circle of fifths (interval power-series of the fifth), whereby we admittedly “land” on the ratio 729:

1/729asˇ101/243esˇ81/81bˇ71/27 f51/9c31/3g2

1 d

3/1aˆ19/1eˆ327/1hˆ481/1fisˆˆ6243/1cisˆˆ8729/1gisˆˆ9

which, reduced by octaves, yields the chromatic scale:

asˇ aˆ bˇ hˆ ccisˆˆ desˇ eˆ ffisˆˆ ggisˆˆ

With tempering of as and gisˆ, we have the “scale of 12 Lu” of Chinese music, and Thimus remarks on this (I, 89) that the system of this scale “moves into those modulatory limits (alterations) beyond which the Classical composers of the 16th century did not go in their application of chromatics.” Or else, we start from the two “real” poles, which for 1/1d have the values 48/15bˆ and 15/48fis,, and construct, upwards or downwards in fourth steps, the series:

81/256fisˆˆ,,27/64hˆ,,9/16eˆ,,3/4aˆ1

1 d

4/3g16/9cˆ′ 64/27fˆ′ 256/81bˇˇ′′

which, after octave transposition, yields the nine-step scale:

aˆ bˇˇ hˆ cˆ d eˆ fˆ fisˆˆ g

These nine steps, with the elimination of b and fis, contain the diatonic (C-major) scale, but precisely these two chromatic steps b and fis were necessary, as Thimus explains (I, 90), to represent the so-called octave-categories of the system of the eight Gregorian church modes within the steps of the natural key-signature (without alteration of the clef), as well as within the so-called musica ficta of the signatures with a sharp and a flat in the Guidonian tone-system. From these analyses, it can already be seen that the I Ching hexagrams are the expression of a general regularity in musical terms.

However, by means of harmonic analysis, we can explore this regularity further. For this, we return once more to the eight primal signs (Koua = primal signs)-see Table 466.

Figure 466

Category I lists the eight signs with their names. They are analyzed according to Chinese usage from bottom to top; we set the whole lines equal to the whole numbers, the broken lines equal to their reciprocals. Categories II to IV attempt to harmonically analyze the individual triads a b c through insertion of various number values (V belongs to IV and simply gives the logarithms of IV). Category II proceeds according to Thimus's scheme, and sets a = 1, b = 2, c = 3 for the straight lines and 1/a = 1/1, 1/b = 1/2, and 1/c = 1/3 for the broken lines. The insertion of tone-values stands under the sign of the fifth intervals g and f (here we return to using 1/1c as generator-tone), the “father” designated as g, the “mother” as f,,,. In the ratio of the “children” an agreement of character of the first son with the mother is shown, a similarity of the second son with the mother, and a similarity of the third son with the father; the first daughter is identical in character with the father, whereas the second is similar to the father and the third is similar to the mother. Regarding the “character identity,” however, it must be remarked that since the Chinese also assign a meaning to the location of every line, e.g. the “identity” of father === and first daughter === is inwardly present (both the value g), but is different in terms of its “material” spatial structure. The formula of the father is 1 · 2 · 3, that of the first daughter is 1/2 · 2 · 3-identical as tone-value, but different in the octave and the character of the lines, as well as their situs = location. This observation should be kept in mind with all further identities.

We come closer to the “family relationship” of these eight signatures through the fifth-analysis of category III. This was well known in Chinese harmonics, and therefore we may use it without scruples. Here, father and mother have the ratio to the generator-tone 1/1c of the minor third and sixth (aˆ) and (esˇ)-whereby we are reminded of the third as the “gender tone.” Son and daughter here emphasize the three identical values: the fifths f and g.

Most interesting, however, is the analysis of category IV. Here I have inserted the first, second, and third powers of the fifths, i.e. the potentiation that is known to us from the Pythagorean scale and that plays such an important role in Plato's Timaeus. It seems likely to me that this was known of and used in ancient Chinese harmonics, since fundamentally it only consists of very simple multiplication. But this is irrelevant to the results of our analysis, since the I Ching diagram gives psychical forms in bar-notation, which we can elicit in other ways with harmonic methods. Here the father- and mother-principle returns in the extreme ratios 729/1fis and 1/729ges, whose logarithms 510 and 490 only differ by 20 units (reduced by octaves), which their enharmonic character shows, although one is very high (heaven-number) and the other very low (earth-number). This number 729 is otherwise almost twice the number of days in the year (2 × 365 = 730) and thus has symbolic-cosmic significance. The progression of the “sons” climbs upwards in the whole-tone steps c bˇ asˇ, the progression of the “daughters” in analogous whole-tone steps c d eˆ. If we arrange the tones of the “family” of eight scalewise, the result is the strange scale:

so, a definite whole-tone scale of completely symmetrical “genealogical” construction: parents in the middle, daughters to the left, sons to the right. Moreover, the intervals of this whole-tone scale emerge from the important steps we know of: the major whole-tone of type 9/8d 170, and the minor whole-tone of nearly 10/9dˇ 152-again, two enharmonic steps-whereas the enharmonic middle interval of the “parents” has 19 log units, an interval that is very close to the so-called synthetic comma of 18 log units (80/81 or 81/80), and thus once again shows its harmonic-rational significance.

I began by investigating all the I Ching hexagrams (from R. Wilhelm's edition) according to the Thimusian scheme (table 466, category II), and calculated the number- and tone-value for every hexagram. The result is that all 64 hexagrams yield only the following ten values in multiple repetitions-here I present them arranged in numerical terms, as well as in musical notation:

Figure 467

This is a chromatic scale without the main steps c f and g. The many “identities” must naturally be observed from the above viewpoint of the various “locations” of the lines; thus all hexagrams of equal harmonic value are still different in themselves.

But the fruitfulness of our analysis of categories IV (and V) in Table 466 allows us to suppose that with this power-analysis we can come closer to finding the inner psychic meaning of the I Ching hexagrams than with the too-simple Thimusian method. Unfortunately, I have not yet found the time to calculate all 64 hexagrams according to this scheme, and therefore can only note the results for the first four (Fig. 468).

Figure 468

1. Twice the “father” becomes the Kiän hexagram, the creator. The total value is his log 020, a step that is enharmonic to the generator-tone c.

2. Twice the “mother” becomes the Kun hexagram, the receiver. The total value is deses log 980, likewise an enharmonic step to c. The generating and receiving principle, as above in the trigrams of “father” and “mother,” is in an enharmonic relationship, albeit not directly in itself, but rather in terms of an imaginary c that creates both, i.e. unity. All remaining values of the other hexagrams will show, as for 3 and 4, total values built from two independent trigram values, i.e. a psychic unity. Even if different hexagrams should exhibit identical tone-number characters on the outside, still the inner structure of their step-building is different. This is somewhat comparable to the tone-locations in the “P” system, where, for example, 2/1c4/2c6/3c etc. are not different in themselves, but are different in the tone-location, i.e. the place in the configuration space.

Those who know and “practice” the I Ching will wonder what has been gained for the understanding of this book with this type of harmonic analysis. Each of its hexagrams has, in any case, a title (see the titles of the first four ones just discussed), each has enough commentaries on its “meaning,” etc. But this is not in “tones,” but in a conceptual speech that we can understand. Through various modifications and combinations of these hexagrams, the imagination and intuition are given full rein.

But let us take ad exemplum only the first two hexagrams, Kiän = the creative and Kun = the receptive. In Wilhelm's edition, much is written about these two concepts, more or less describing and interpreting the concepts of the “creative” and the “receptive,” furnishing them with rules and modes of conduct, etc. The harmonic analysis, however, gives the reasons, in a precise and concrete way, for why in these signs the “creative” is connected with the “heavens,” the great light with the “masculine,” etc.; why the “receptive” is connected with the “earth,” “small,” “dark,” “feminine,” etc.: in the trigrams, the appearance of the “gender interval,” the third; the peculiar enharmonic equivalence of both principles; that they are “of one flesh” yet different in essence; the assignment of the undivided lines to 1/1 → ∞/1, the endlessly large, the light, heaven, and the divided lines 1/∞ ←1/1 to the limited small, dark, receptive; the remarkable number 729 and its closeness to twice the number of days in the year. In contrast to the non-committalness, the stark claims (whose antiquity is indeed honorable, but tells us nothing about the effective truth content) of the I Ching commentators, harmonic analysis thus reveals the reasons and leads these reasons back to a value-emphasized number-system, from which views of other domains are possible.

Naturally, a sensitivity to a differentiated tone- and interval-perception, a thoroughly trained psychical grasp of the characteristic steps through the ear (in monochord experiments), and spiritual insight into the symbolism of these steps that emerges from this, are central to harmonic analyses of this type. Anyone who inherently has, or has developed, an ear for this will also be able to judge whether the analyses of the further 64 I Ching diagrams show similar results. It is this sensitivity, this “sixth sense” of akróasis, that lies at the root of all harmonic investigations, thoughts, and perceptions, and I can imagine that in future times, there may be an ingenious harmonist who will concentrate the basics of harmonics in a book similar to the I Ching.

§50.9. Advice for the Harmonic Analysis of Number Symbols(cf. §50.4)

This chapter in itself far exceeds the space permitted by the limits of a textbook. But the domain of number symbolism itself is enormously large, and treated (and here very meagerly) in the strict scientific sense as something simply “historical,” “folkloric,” etc., while, as is shown by the above examples and all the work of A. von Thimus, there is still an almost inexhaustible field of harmonic research at our disposal. Therefore, I thought I should show a few concrete examples of how and by what means harmonic analysis should proceed here. The “uninitiated” will admittedly find the analyses of the I Ching diagrams very complicated; but anyone who has solidly worked through this book up to this point will be impressed by the simplicity of these investigations.

We have seen not only that the tone-numbers, intervals, selections such as scales, interval powers, etc. can and should be used for the analysis of number symbolism, but that also the harmonic configuration space (“P” and its modifications) must be taken into consideration. In my essay on Pythagoras, for example, the interpretation of the Pythagorean number-cosmos was only possible by means of analogies with the polar “P”, which however hit the target. One should carefully note the possible approaches from case to case, and not give oneself up to haphazard tone-analysis. In many cases, harmonics will fail, and then it is better to be content with a negative result than to use harmonics like a straitjacket. With the relics of the ancient number-harmonics, we must not forget-as Thimus noted again and again regarding the ancient sources-that in ancient wisdom, especially in its Gnostic form, much was kept deliberately secret, and thus often misunderstood and wrongly recorded later. The first task, then, is to separate the wheat from the chaff, as far as possible; only then will harmonic analysis have meaning and purpose.

§50.10. “Life-Numbers”

Since the belief in a significance of certain “life-numbers” is widespread, and certain temporal-spatial periodicities and rhythms are also not to be repudiated, I will give a simple overview here of how one must proceed with the harmonic analysis of such “numbers.” The method has already been mentioned in my Abhandlungen, pp. 49-50.

One should draw up a table similar to Fig. 469, which is simply an “overtone series” dissolved into interval constants and powers-naturally those who hope to live 100 years can lengthen it as much as they wish beyond the ratio 81.

Suppose we analyze the dates of our own lives from the viewpoint of this scheme, first purely biologically. It may appear that the individual spaces of the interval constants are equal in value amongst themselves, despite respective doubling of their periods. What we biologically “experience” between the first and second year is thus equal in value to what we experience between the second and the fourth, the fourth and the eighth, the eighth and sixteenth, etc. The same goes for the fifth-octaves 3, 6, 12 ..., the third-octaves 5, 10, 20, etc. A closer examination will also ascribe certain intervals to periods of a definite type, for example, the seventh-octave 7, 14, 28, 56 to a sexual type, etc. But not only the biological, also the psychical and spiritual “stations” of our life appear to consolidate in these harmonic “numbers.” I myself have found peculiar equivalents with regard to important stages of life. There are apparently “octave-,” “third-,” etc. breaks, which are connected with important changes in our fate, or which provoke them. Any reader can easily perform his own analyses here, on himself, his family, etc. No extensive investigations of this type have yet been made. One can further imagine that some important event in life also makes the relevant life-number “independent,” i.e. changes it to 1/1 and from there on the intervals follow autonomously. Furthermore, the predominant intervallic character of the life in question could be retrospectively defined and studied from the data obtained; for example, the relevant intervals within a family, a line of ancestors, or simply between two or more different life-number successions could be consonant or dissonant. These are no Utopias; but the investigation of such connections requires particular assiduousness and strict disavowal of all “augury,” since one can never know or say with precision to what extent, for example, a certain life-number is autonomous and radiates out into the future. But for insight into a biological-psychic arrangement of every life's fate, this method appears to me from my own experience to be retrospectively important. (The oldest testimony to the significance of the number 7 in human stages of life is in the elegy of the “Hebdomad” of Solon. He prescribes at age 7 the second teeth, at 14 puberty, at 21 beard growth, at 28 the maximum bodily strength, at 35 the time to marry and have children, at 42 the full formation of character, at 49 and 56, thus through 2 hebdomads, the mature development of understanding and discourse, in the ninth hebdomad their deterioration, and in the tenth, preparation for the end.)